Problem: A composite function can be written as $w\bigl(u(x)\bigr)$, where $u$ and $w$ are basic functions. Is $g(x)=\sqrt{\ln(x)}$ a composite function? If so, what are $u$ and $w$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $g$ is composite. $u(x)=\sqrt{x}$ and $w(x)=\ln(x)$. (Choice B) B $g$ is composite. $u(x)=\ln(x)$ and $w(x)=\sqrt{x}$. (Choice C) C $g$ is not a composite function.
Explanation: Composite and combined functions A composite function is where we make the output from one function, in this case $u$, the input for another function, in this case $w$. We can also combine functions using arithmetic operations, but such a combination is not considered a composite function. The inner function The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function. Here, we have $\ln(x)$ inside the radical. We evaluate this expression first, so $u(x)=\ln(x)$ is the inner function. The outer function Then we take the square root of the entire output of $u$. So $w(x)=\sqrt{x}$ is the outer function. Answer $g$ is composite. $u(x)=\ln(x)$ and $w(x)=\sqrt{x}$. Note that there are other valid ways to decompose $g$, especially into more complicated functions.